Factor models with many assets: strong factors, weak factors, and the two-pass procedure

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1 Facor models wh many asses: srong facors, weak facors, and he wo-pass procedure Sanslav Anaolyev CERGE-EI and ES Anna Mkusheva MI Augus 07 PRELIMIARY AD ICOMPLEE. PLEASE, DO O DISRIBUE! Absrac hs paper re-examnes he problem of esmang rsk prema n facor prcng models. A ypcal feaure of daa used n he emprcal leraure o esmae such models s he presence of weak facors ha are prced and, a he same me, he presence of unaccouned srong cross-seconal dependence n he errors. Anoher feaure of ypcally used daa s moderaely hgh cross-seconal dmensonaly. Usng an asympoc framework where he number of asse/porfolos grows proporonaely wh he me span of he daa whle he rsk exposures of weak facors are local-ozero, we show ha under such crcumsances he convenonal wo-pass esmaon procedure delvers nconssen esmaes of he rsk prema. We propose a modfed wo-pass procedure based on sample-splng nsrumenal varables esmaon a he second pass. he proposed esmaor of rsk prema s robus o he presence of srong unaccouned cross-seconal error dependence, as well as o he presence of ncluded facors ha are prced bu weak. We derve he many-asse weak facor asympoc dsrbuon of he proposed esmaor, show how o consruc s sandard errors, verfy s performance n smulaons, and apply o ofen-used daases from exsng emprcal sudes. Keywords: facor models, prce of rsk, rsk prema, wo-pass procedure, srong facors, weak facors, dmensonaly asympocs, weak facor asympocs. JEL classfcaon codes: C33, C38, C58, G. Address: Sanslav Anaolyev, CERGE-EI, Polckych vězňů 7, Prague, Czech Republc; e-mal sanslav.anaolyev@cerge-e.cz. Address: Deparmen of Economcs, M.I.., 50 Memoral Drve, Buldng E5, Cambrdge, MA, 04, USA; e-mal amkushe@m.edu. We are graeful o Davd Hughes for research asssance.

2 Inroducon Snce he nroducon of CAPM by Sharpe 964 and Lnner 965, lnear facor prcng models have grown no a very popular sub-feld n asse prcng. Harvey, Lu and Zhu 06 ls hundreds of papers proposng, jusfyng and esmang varous facor prcng models. A ypcal paper n hs area proposes a small se n he range of 3-5 of observed rsk facors. he classcal facor s he marke porfolo excess reurn, whch was augmened by he sze facor SMB small-mnus-bg reurn and he book-o-marke facor HML hgh-mnus-low reurn n Fama and French 993. Among oher well-known prcng facors are he momenum facor MOM Jegadeesh and man, 993 and he consumpon-o-wealh rao cay Leau and Ludvgson, 00. radonally, he model s esmaed usng wha s commonly known as he wo-pass esmaon procedure Fama and MacBeh, 973; Shanken, 99. hs procedure, however, reles on an dealsc seup wh srong denfcaon of rsk prema. Emprcally, n realsc crcumsances such condons ofen do no hold. For example, he recen leraure shows ha he msmeasuremen of he rue rsk facors leads o weakness n he observed facors and srong cross-seconal dependence n he errors Klebergen and Zhan 05, whch may resul n all sors of dsorons n esmaon and nference n heory and her non-relably n pracce e.g., Kan and Zhang 999, Andrews 005, Klebergen 009. Recen papers by Kan and Zhang 999, Klebergen 009, Bryzgalova 05, Burnsde 05, Gospodnov, Kan and Robo 06 all pon ou ha rsk exposures or beas o some observed facors end o be small o such an exen ha he her esmaon errors are of he same order of magnude as he beas hemselves. he observed phenomenon s very smlar o he wdely suded weak nsrumens problem. he remedes for some of hese falures proposed n he leraure rely eher on complcaed nference ools robus o weak denfcaon Klebergen 009, or requre he use of dmenson-reducon echnques Bryzgalova 05. Along wh a combnaon of problems of small beas and mssng facors, we also consder one very mporan emprcal feaure of ypcally employed daases he presence of a large number of asses or porfolos ofen comparable o he number of perods over whch reurns are observed. We consder an asympoc framework where he number of asses/porfolos of whch reurns daa s used for esmaon grows wh s me-seres Somemes he wo-pass procedure s referred o he Fama-MacBeh procedure Fama and MacBeh, 973. See Cochrane 00, secon.3 on her numercal equvalence when beas are me nvaran. he mehod of obanng vald sandard errors ha accoun for he wo sep naure of he procedure was gven n Shanken 99.

3 dmenson. Such dmenson asympocs s lkely o provde a more accurae asympoc approxmaon o he fne sample properes of esmaors and ess. As s known from he concepually-relaed leraure on many nsrumens Bekker, 994 and many regressors Anaolyev, 0, large dsorons n convenonal esmaon and nference may arse when he number of nsrumens and/or regressors ceases o be a ny fracon of he sample sze, and correcons of esmaors and ess requre a framework wh he dmenson asympocs whn whch hese fracons are asympocally non-zero. he many-asse asympoc framework was already ulzed prevously by Gaglardn, Ossola and Scalle 06. Whn he new dmensonaly asympocs we show ha he presence of small beas leads o a falure of he classcal wo-pass procedure, whle he addonal presence of mssng facors exacerbaes he problem. We propose economerc procedures ha are robus o boh of he ssues wh facors he weakness of observed facors and he presence of unobserved facors n he errors, and, n conras o he remedes proposed elsewhere, are easly mplemenable usng sandard regresson ools n parcular, nsrumenal varables regressons and wo-sage leas squares. he esmaors we propose are conssen and asympocally mxed gaussan; moreover, usng he varance esmaors whose consrucon we descrbe, he sandard nference ools such as - and Wald ess can be appled n a convenonal way. Our usage of dmensonaly asympocs has an mporan mplcaon: even hough we assume ha some observed facors have small beas, does no lead o a weak denfcaon problem as currenly defned n he leraure, and we can oban a conssen esmaor. hs dsncon s smlar o he one beween he leraure on weak nsrumens and he leraure on many weak nsrumens e.g., Hansen, Hausman and ewey, 008. In he laer, he weakness of nsrumens mples denfcaon f hese nsrumens are numerous and, whle he classcal wo sage leas squares SLS esmaor has large dsorons, one can consruc a conssen and asympocally gaussan esmaon procedure. Our new esmaon approach uses some deas from he many-weak-nsrumens leraure such as sample-splng and he use of nsrumenal varables esmaon a he second sep. However, we addonally face and solve a dsnc problem ha one does no encouner n he many weak nsrumen leraure, namely, ha of he presence of an unobserved facor srucure n he error erms ha creaes srong cross-seconal dependence n he panel of reurns and s smlar o he classcal omed-varables problem n he second-pass regresson of he wo-pass procedure. he paper s organzed as follows. Secon nroduces noaon, dscusses he rel-

4 evance of our asympoc approach, and argues for he presence of a sgnfcan facor srucure n he errors. Secon 3 nroduces and dscusses echncal assumpons. Secon 4 explans he asympoc falure of he classcal wo-pass procedure and provdes dealed nuon as o why ha happens. We propose our esmaon mehod n Secon 5, descrbe wha movaes and explan why works. We also sae he formal heorem on he conssency of he newly-proposed 4-spl esmaor. Secon 6 s devoed o dervng nference procedures ha use our 4-spl esmaor, n parcular, we show he asympoc valdy of a properly consruced Wald es. he resuls of numerous smulaons suppor our heorecal fndng and are placed n Secon 7. Secon 8 revss some promnen emprcal applcaons of he facor-prcng model. A word on noaon: 0 l,m says for a zero marx of sze l m, I m s an m m deny marx, for an m l A marx A and an m l B marx B symbol A; B says for he m l A + l B marx obaned from placng he nal marces sde-o-sde. Formulaon of he problem As menoned n he Inroducon, a paper n he area of fnancal facor models ypcally proposes a small se of observed rsk facors descrbed by a k F vecor F, wh a usually small dmenson k F. An asse or porfolo of asses wh excess reurn r has exposure o several rsk facors, whch s quanfed by he asse s beas β = varf covf, r. A ypcal clam n lnear facor-prcng heory s ha exposure o rsk beas fully deermnes he asses expeced reurns. Parcularly, here exss a k F -dmensonal vecor of rsk prema λ such ha Er = λ β. he lnear facor-prcng model s equvalen o he followng formulaon: r = λ β + F EF β + ε, where he random error erms ε have mean zero and are uncorrelaed wh F. One specal case ofen menoned n he leraure occurs when he F facors are asse reurns hemselves and are supposed o be prced by he same model; n hs case, heorecally λ = EF. We wll no make hs assumpon and wll consder he general case when λ may dffer from EF. wo-pass procedure he esmaon and formaon of nferences on rsk prces, λ, are usually accomplshed by a procedure commonly known as he wo-pass esmaon procedure Fama and MacBeh, 973; Shanken, 99, whch s appled o a daa se conssng 3

5 of a panel of asse excess reurns {r, =,...,, =,..., } and of observaons of realzed facors {F, =,..., }. As a frs sep, one esmaes β by runnng a me seres OLS regresson of r on a consan and on F for each =,...,. he second sep produces an esmae of λ denoe λ P by regressng he me-average excess reurn = r on he frs-sep esmaes, β. Under suable condons, λ P s proved o be boh conssen and asympocally gaussan. Dscussons of he sascal properes of he wo-pass procedure appear n Fama and MacBeh 973, Shanken 99, and Chaper of Cochrane 00. Recenly, several promnen researchers have rased he concern ha he wo-pass procedure may provde msleadng esmaes of rsk prema; see, for example, Kan and Zhang 999, Klebergen 009, Bryzgalova 05, Burnsde 05, Gospodnov, Kan and Robo 06. hey surmse ha he reason for hese erroneous nferences s aached o he emprcal observaon ha eher some column of β = β,..., β s close o zero, or, more generally, he k F marx β appears close o one of reduced rank less han k F for a majory of well-known lnear facor-prcng models. he observed phenomenon s smlar o he wdely suded weak-nsrumens problem Sager and Sock, 998: f some of he observed facors F are only weakly correlaed wh all of he reurns n he daa se, hen he nose ha arses n he frs-sage esmaes of he correspondng componens of β wll domnae he sgnal, and he second-pass esmae of he rsk prema λ wll be over-sensve o small varaons n he sample. In order o model he observed phenomenon, Klebergen 009, Bryzgalova 05 and Gospodnov, Kan and Robo 06, among ohers, consdered a drfng-parameers framework n whch hey model some componen of β o be of order O assumng ha he number of me perods,, ncreases o nfny, whle he number of asses,, says fxed. In such a seng he frs-sep esmaon error s of order of magnude O p, whch s comparable o he sze of he coeffcens hemselves. he wo-pass procedure could be reformulaed as an IV-ype esmaon, and hen he above-descrbed drfng-sequence asympocs characerze he weak-nsrumens case. hs framework mples nconssency of he wo-pass esmaes for he rsk premum on small componens, poor coverage of regular confdence ses even for he rsk premum of srong facors, and asympoc nvaldy of classcal specfcaon ess and ess of hypoheses abou rsk prema. Followng hs radon and acknowledgng emprcal evdence provded n Klebergen and Zhan 05 and Bryzgalova 05 we also make use of drfng-parameer modelng. We assume ha he k F vecor of facors F can be dvded no wo subvecors: a 4

6 k dmensonal vecor F, and a k vecor F, here k F = k + k such ha he rsk exposure β, o facor F, wll be srong, whle he rsk exposure coeffcens β, o facor F, wll be drfng o zero a speed. We make hese order assumpons for rsk exposure more accurae n he nex secon. he mos mporan feaure of our modelng wll be ha he sandard error of he frs-sep esmaor of β, wll be of he same order of magnude as he coeffcen self. A more general reamen of he neardegenerae rank condon consders some k -dmensonal lnear combnaon of facors unknown o he researcher o have a local-o-zero of order O exposure coeffcen, whle he exposure o rsk formed by he orhogonal k -dmensonal lnear combnaon remans fxed. All our resuls are easly generalzable o hs seng, as we do no assume he researcher knows whch facors or combnaon of facors bear small coeffcens of exposure. However, o smplfy he exposon we wll sck o he dvson of facors no wo sub-vecors. hs paper devaes from he prevous leraure n wo drecons. Frs, we consder an asympoc seng where boh and grow o nfny. We noce ha n many common daa ses researchers use n he esmaon of facor prcng models, he number of asses,, s large when compared o he number of me perods. he celebraed Fama-French daa se provdes reurns on = 5 sored porfolos for abou = 00 perods. he ofen-used Jagannahan-Wang daa se Jagannahan and Wang 996 conans observaons on = 00 porfolos observed for = 330 perods. Leau and Ludvgson 00 use Fama-French = 5 porfolos, he reurns for whch are observed durng = 4 quarers. Gaglardn, Ossola and Scalle 06 use = 44 ndusry porfolos observed durng = 546 monhs. In hese cases s hard o beleve ha he asympoc resuls derved under he assumpon ha s fxed would provde an accurae approxmaon of fne-sample dsrbuons. Indeed, among oher hngs, Klebergen 009 dscovers ha he bas of he wo-pass esmae of rsk prema s srongly and posvely relaed o he number of asses f he oal facor srengh s kep consan. In hs paper we consder asympocs when boh and ncrease o nfny whou resrcng he relave growh beween hem. he second and man devaon of hs paper from he exsng leraure s our explc acknowledgmen of hgh cross-seconal dependence among error erms ε n model. In parcular, we assume ha errors have a facor srucure. amely, hs means ha here exss a mssng unknown and unobserved o he researcher facor v and loadngs µ such ha ε = v µ + e, 5

7 where he clean errors e are only weakly cross-seconally dependen o such an exen ha asympocally we may gnore her dependence exac formulaon of hs assumpon appears n he nex secon. Smlar weak-dependence assumpons appear n approxmae facor models e.g., Ba and g 00. he assumpons on loadngs µ guaranee ha he facor srucure wll be srong enough o be boh deeced emprcally and asympocally mporan for nferences. An nsghful dscusson of weak vs srong facor srucure and cross-seconal dependence appears n Onask 0. Below we provde wo heorecal reasons as o why we expec facor srucure n many lnear facor-prcng models. hen we pon o emprcal evdence ha a mssng facor srucure s ndeed presen n some well-known facor-prcng models. Example. If one does no observe he rue rsk facors ha prce asses bu only proxes for hem, hs would lead o a facor srucure n errors see Klebergen and Zhan 05, who show n parcular ha hs furher leads o spurously large values of secondpass R. Assume for a momen ha he marke s prced by rsk prema on rsk facors G. For exposonal smplcy we assume ha r = G β G + ε G, where he shocks ε G are drawn wh mean zero and fne varance ndependenly crossseconally and me-seres and ndependen from G. Also assume ha G s saonary wh varance Σ G. Assume ha he economercan does no observe G, bu raher has a proxy for, F = α + δg + ϵ, where ϵ has mean zero and s uncorrelaed wh G and shocks ε G. For example, ϵ may sand for a measuremen error or conamnaon by oher macro varables no conneced o asse prces. Denoe he varance marx of ϵ by Σ ϵ. If δ s a full-rank square marx, hen one can show ha proxes F can prce asses as well as he rue rsk facors G. Indeed, β = V arf CovF, r = δσ G δ + Σ ϵ δσ G β G = Aβ G, where he marx A = δσ G δ + Σ ϵ δσ G s a full-rank square marx. hus, Er = EG β G = λβ, where λ = A EG. So we see ha f he economercan s ryng o esmae a lnear facor-prcng model usng facors F, she has a correcly-specfed model; however, hs 6

8 model unlke he model wh he observed facors G has a facor srucure n s error erms. Indeed, usng some smple algebra one can show ha equaon holds wh ε = Σ ϵ δ Σ G G EG ϵ β + ε G = v µ + ε G. Wha s neresng here s ha whle he facors v and he errors ε hemselves are uncorrelaed wh he observed facors F, he loadngs on he error facors, µ s, and he orgnal loadngs, β s, are closely relaed n hs parcular case µ = β. We wll make use of hs observaon n our dscusson of he valdy of he wo-pass procedure. Example. Consder a suaon n whch one of he rsk facors drvng asse reurns s fully arbraged and hus carres a rsk prema of zero. If an economercan does no observe hs facor bu does have observaons on all oher relevan rsk facors, hen her lnear facor-prcng model ha oms he arbraged facor may sll be correcly specfed, whle he arbraged facor s moved o he error erm, resulng a mssng-facor srucure o he errors. Klebergen and Zhan 05 provde numerous peces of emprcal evdence ha resduals from many well-known esmaed lnear facor-prcng models have non-rval facor srucures. For example, hey pon ou ha he frs hree prncple componens of he resduals from dfferen prcng-model specfcaons used n he promnen paper by Leau and Ludvgson 00 explan somewhere from 8% o 95% of all resdual varaon Klebergen and Zhan 05, able 3. hey also show ha he larges egenvalue of he covarance marx of resduals n all hese examples s very large and srongly separaed from oher egenvalues ha are bunched ogeher. Combnng hese resuls on he larges egenvalues of he resdual covarance marx wh he heorecal resuls on he lmng dsrbuon of s egenvalues from Onask 0, one would suspec here s a leas one srong facor presen n he resduals. A leas fve oher promnen facor-prcng sudes ced n Klebergen and Zhan 05 demonsrae smlar evdence of a srong facor srucure lef n he resduals. Relaon beween facor srucure and correc specfcaon. One may wonder wheher he fac ha he errors ε n he model have a facor srucure mples ha he prcng model s msspecfed. he answer s o ; he lnear facor-prcng model descrbes he expecaons of excess reurns, whle he facor srucure n he errors s relaed o he covarances or co-movemens of he asses reurns. I s easy o see ha f he rsk exposure and rsk prema on he varables F prce he asses, hen he varables 7

9 F co-move he asses reurns and produce facor-srucure dependence n he reurns. However, no all co-movemens of reurns mus carry non-zero rsk prema; hose comovemens can be placed n he error erm whou causng msspecfcaon of he prcng model. In hs paper we assume ha he correc specfcaon of a prcng model requres keepng n he model hose prcng facors F, ha carry small coeffcens of exposure β, and produce only a weak facor srucure n reurns. We show ha droppng such observed facors from he specfcaon as opposed o wha s proposed n Bryzgalova 05 leads o asympocally msleadng nferences for boh he wo-pass procedure and our proposed procedure. Our mehod s robus o mnor msspecfcaons ha allow one o drop hose prcng facors ha carry loadngs of order o ; he exac formulaon appears n Secon Seup and assumpons 3. Model We consder he problem of esmaon and nference on he rsk prema λ based on observaons of reurns {r, =,...,, =,..., } and facors {F, =,..., } comng from a correcly specfed facor-prcng model: r = λ β + F EF β + v µ + e, where he correc specfcaon means ha he random unobserved facor v has zero mean and s uncorrelaed wh F, he dosyncrac error erms e also have zero mean and are uncorrelaed wh F. We also assume ha hey are uncorrelaed wh v. Denoe F o represen he sgma-algebra generaed by he random varables F,..., F and v,..., v ; le γ = β, β, µ, and Γ = γ,..., γ. 3. Assumpons We make he followng assumpons. Assumpon FACORS. he k F vecor of observed facors F s saonary wh fne fourh momens, a full-rank covarance marx Σ F, and summable auo-covarances. he k v vecor of unobserved facors v s such ha he followng asympoc saemens 8

10 hold smulaneously: F EF 0, Ω F ; = η = = η v, = Σ F F v η; v η v 0 kv,, I kv, where vecη 0 kf k v,, Ω vf and F = F s= F s. = Assumpon LOADIGS. As boh and ncrease o nfny, we have Γ Γ Γ, where Γ s a posve-defne k k marx wh k = k F + k v. In addon we assume ha max, = γ 4 <. We adop he followng noaon: Γ β µ s he k k µ sub-block of marx Γ correspondng o he lm of = β, µ. Oher sub-marces are denoed smlarly. Assumpons ERRORS. Condonal on F, he random vecors e = e,..., e are serally ndependen, and Ee F = 0 for all. Le ρ, s = = e e s. hen, sup sup s E [ + F 4 ρs, + ] < C. Le S = = e. hen, = F S = o p and = F F S p Σ SF. v Le W = = γ e. hen, E [ + F W ] <. 3.3 Dscusson of Assumpons Assumpons FACORS. As a par of he error erm, v s uncorrelaed wh F. One can come up wh a varey of assumpons on decayng dependence and momen condons ha would guaranee some Cenral Lm heorem saed n and. he resrcon ha he asympoc covarance marx be he deny marx s jus normalzaon, as neher v nor loadngs µ are observed. 9

11 Assumpon LOADIGS. In hs paper we rea he loadngs β and µ as unknown consan non-random vecors he rue values of whch may change wh he sample szes and, whch s an example of he so-called drfng parameers asympocs. Assumpon LOADIGS characerzes he sze of he loadngs as he sample sze ncreases. oce ha he loadngs on he facors F, and v are reaed dfferenly han he loadngs on F, ; followng Onask 0 we wll refer o he former as srong facors and he laer as weak facors. he cross-seconal average of squared loadngs s closely conneced o he explanaory power he facors exhb n cross-seconal varaon. he assumpons we make on he loadngs β, and µ guaranee ha he explanaory power of he facors F, and v domnaes ha of he dosyncrac error erms. he average squared loadng on he facor F,, however, converges o zero a a rae of / ; f and ncrease proporonally, hs wll lead o facor F, havng explanaory power comparable o ha of he dosyncrac errors. One characersc of a weak facor s he followng: f had no been observed we could neher have conssenly esmaed va he mehod of prncple componens appled o he esmaed cross-seconal covarance marx nor conssenly deeced. he loadngs β, are asympocally of he same order of magnude as β, dvded by. Assumpon LOADIGS makes he sandard devaon of he frs-sep esmae β, of he same order of magnude as β, self. As we show below, hs s enough o make he wo-pass esmaor of he rsk prema λ on he weak facor F, nconssen and o nvaldae he classcal confdence nerval for he rsk prema λ on he srong facor F,. he modelng assumpon ha makes β, drf o zero a he / rae s smlar o assumpons Klebergen 009, Bryzgalova 05 and Gospodnov, Kan and Robo 06 make. In hese papers he auhors assume ha remans fxed, whch makes he asse prema λ a weakly-denfed parameer. We, however, assume ha ncreases o nfny whch, ogeher wh hs assumpon, allows one o consruc a conssen esmaor for λ, he wo-pass esmaor sll beng nconssen. hus, our seng s no a case of weak denfcaon. I s also mporan ha he assumpon on loadngs µ be such ha he unobserved facor v n he error erms s srong. hs s conssen wh he emprcal observaons Klebergen and Zhan 05 presen. hs also guaranees ha he presence of he facor srucure plays an mporan role n he asympocs of wo-pass esmaon. he error erms may also have weak facor srucure; we do no explcly specfy hs because wll no have asympoc mporance for he esmaon procedures we consder here. 0

12 Assumpon ERRORS. Assumpons ERRORS are hgh-level assumpons he man goal of whch s o allow very flexble weak cross-seconal dependence among he dosyncrac errors, as well as flexble condonal heeroscedascy and dependence n hgher-order momens of errors and facors. he random varables ρs, sand for a normalzed emprcal analog of he error auocorrelaon coeffcen, S s an emprcal varance, and W s a normalzed weghed average error. hese varables are normalzed so ha hey are sochascally bounded when he errors are cross-seconally..d. Seral ndependence of errors as saed n Assumpon ERRORS s conssen wh he effcen marke hypohess and he unpredcably of asse reurns; and s generally conssen wh emprcal evdence and he radon n he leraure. hs assumpon may be weakened, hough we do no pursue hs n he curren paper. In order o undersand Assumpons ERRORS we provde below a se of more resrcve prmve assumpons ha are common n he leraure and ha guaranee he valdy of our hgh-level Assumpons ERRORS. We also provde an emprcally relevan example no covered by he prmve assumpons below bu ha sasfes he hgh-level Assumpons ERRORS. Assumpons ERRORS he facors {F, =,..., } are ndependen from errors {e, =,...,, =,..., }; he error erms e = e,..., e are serally ndependen and dencally dsrbued for dfferen wh Ee = 0 and sup, Ee 4 <. Le E, = E [e e ] be he covarance marx when he sample sze s and n cross-secon and me drecons, correspondngly. For some posve consans a, c and C, c < lm nf mn eval E, < lm sup max eval E, < C,, and lm, re, = a. E = e Ee < C., Lemma Assumpons LOADIGS and Assumpons ERRORS ERRORS. mply Assumpons he prmve Assumpons ERRORS are very close o he sandard ones n he leraure. umerous papers ha esablsh nferences n facor models, commonly assume

13 ha he se of varables {F, =,..., } s ndependen from he se {e, =,...,, =,..., }, hough whn-group dependence s allowed; see, for example, Assumpon D n Ba and g 006. Many papers allow for boh me-seres and cross-seconal error dependence. We exclude me-seres dependence whch s jusfed by he effcen-marke hypohess n our applcaon. Assumpon ERRORS s nended o mpose only weak dependence cross-seconally as expressed by he covarance marx; smlar assumpons appear n Onask 0, and a sronger form s used n Ba and g 006. Our hgh-level Assumpons ERRORS are much more general han he more sandard prmve Assumpons ERRORS. In parcular, our assumpons allow for very flexble condonal heeroscedascy n he error erms and me-varyng cross-seconal dependence, whch seems relevan when we consder observed facors ha characerze marke condons lke he momenum facor. Consder he followng example. Example 3. Assume ha errors e have he followng weak unobserved facor srucure: e = π w + η, where w, F s saonary, w s a k w serally ndependen, condonal on F, mes seres wh Ew F = 0 and Ew w = I kw whch s an nnocuous normalzaon as he facor srucure s no observed. Assume E [F 4 + w 4 + ] <. We assume ha he loadngs sasfy he condon = π π Γ π he facors w are weak, and / = π γ Γ πγ. Assume ha he random varables η are ndependen boh crossseconally and across me, are ndependen from w and F, have mean zero and fne fourh momens and varances σ ha are bounded above and such ha = σ σ. As proven n he Appendx, hs example sasfes Assumpons ERRORS. An neresng feaure of hs example s ha allows he errors o be weakly crossseconally dependen o he exen ha hey may possess a weak facor srucure. Moreover, hs facor srucure may be closely relaed o he observed facors F, whch causes he cross-seconal dependence among he errors e o change wh he observed facors F and allows a very flexble form of condonal heeroskedascy. Indeed, he condonal cross-seconal covarance s Ee e j F = π Ew w Fπ j + I {=j} σ. Snce we do no resrc Ew w F beyond he proper momen condons, he srengh of any cross-seconal dependence as well as error varances may change sochascally

14 dependng on he realzaons of he observed facors. hs flexbly s exremely relevan for such observed facors as he momenum. For example, one may consder w = ς gf, F,..., where ς 0, s ndependen from all oher varables; hen for a proper choce of he funcon g one may observe hgher volaly and cross-seconal dependence of he dosyncrac error for hgher values of he observed facor F. 4 Asympoc properes of he wo-pass procedure In hs secon we derve a resul concernng he asympoc properes of he classcal wopass procedure n dfferen models ha may or may no nclude weak observed facors and may or may no have srong mssng facors n errors. Le us nroduce he followng noaon: λ = λ + where F = F s= F s. F EF, = u = = Σ F F e, ow le us nroduce wo asympocally mporan erms, he meanng and he names of whch wll be explaned n he dscusson followng heorem. he frs erm we call aenuaon bas : AB = β β = whle he second s known as omed varable bas = u u λ, OV B = β β = = β µ η v, η λ. hese erms are no bases n an exac sense as hey are random, bu raher hey are sample analogues of he expressons ha are classcally called aenuaon and omed varable bases. oce ha boh quanes are nfeasble hey canno be calculaed from he daa alone as hey depend on unobserved errors e, unobserved facors v and unknown parameers λ and µ. Boh erms are k F vecors. Le AB and OV B denoe k sub-vecor conssng of he frs k componens, whle AB and OV B are k sub-vecors of he las k componens of AB and OV B correspondngly. heorem Assume ha he sample {r, =,...,, =,..., } and {F, =,..., } comes from a daa-generang process ha sasfes he facor-prcng model and assumpons FACORS, LOADIGS and ERRORS. Le λ P denoe he esmae obaned 3

15 va he convenonal wo-pass procedure. Le boh and ncrease o nfny whou resrcons on relave raes. hen he followng asympoc saemens hold smulaneously: OV B I OV B kβ ; ηγi kβ ; η + I k Σ u I k Γβµ + ηγ µµ η v η λ, AB I AB kβ ; ηγi kβ ; η + I k Σ u I k Ik Σ u λ, λ λ 0, ΩF, and λ P, λ AB OV B λ P, λ AB OV B = O p, where Σ u = lm, = u u, wh he las convergence beng esablshed n Lemma 0k,k 4 n he Appendx, I k = 0 k,k s a k 0 k,k I F k F marx, η = I k η s a k F k v k random marx wh η descrbed n Assumpons FACORS, and Ω F s he long-run varance of F. heorem saes he rae of convergence for dfferen pars of he wo-pass esmaor. oce ha he heorem does no mpose a relave rae of ncrease beween and as long as boh ncrease o nfny smulaneously. One observaon s ha he wo-pass procedure canno esmae λ a a rae faser han despe he fac ha he daase has observaons of porfolo excess reurns, and one could expec he rae. hs comes from he fac ha he correc specfcaon mples ha he excess reurns sasfy equaon, whch, f averaged across me, gves: r = λβ + ε. 3 hus, even f β were known, he rue coeffcen λ n he only deal regresson we have ha s, regresson of average reurn on β dffers from he parameer λ we wan o esmae, by he erm = F EF, whch, f mulpled by, s asympocally zero mean gaussan wh varance Ω F. oce ha f all observed facors F are excess reurns hemselves and are assumed o be prced by he same prcng model, hen he Asse Prcng heory provdes an alernave way of esmang rsk prema. amely n such a case λ = EF, and he alernave esmae λ = = F = λ. However, hs esmae s no vald f facors hemselves are no excess reurns or are no prced by he same model. 4

16 oce also ha f he lms of he normalzed OV B and AB are non-zero, hen hese erms ogeher wh λ asympocally domnae he esmaon. hree cases covered by heorem. Below we consder he frs one s he case wh no weak observed facors k = 0. In hs case he heorem delvers he valdy of he wo-pass procedure, namely, he wo-pass esmaor s conssen and asympocally mean-zero Gaussan. For he oher wo more emprcally relevan cases one wh weak observed facors bu no mssng facors, he oher wh weak observed facors and mssng srong facors he wo-pass procedure fals. he wo-pass esmaes of he rsk prema on weak facors are nconssen. he wo-pass esmae of he rsk prema on he srong observed facor s conssen, bu has a bas whch s of he same order of magnude as s sandard devaon. hs nvaldaes all sandard wo-pass nferences n hese wo cases. 4. Case wh no weak observed facors Corollary Assume ha he samples {r, =,...,, =,..., } and {F, =,..., } come from a daa-generang process ha sasfes he facor-prcng model and assumpons FACORS, LOADIGS and ERRORS wh k = 0 no weak observed facors. hen λ P λ Γ ββ Γ βµη v η λ + lm λ λ, where he lm of he rgh-hand-sde s asympocally gaussan wh mean zero. If, n addon o ha, here are no srong mssng facors n errors ha s, µ = 0, hen λ P λ = λ λ + o p 0, Ω F. hs s a posve saemen abou he wo-pass procedure, whch clams ha f all observed facors are srong, hen he wo-pass procedure s -conssen and provdes asympocally mean-zero gaussan esmae when boh,. If he error erms have a srong facor srucure, does no lead o a bas bu may ncrease he asympoc varance. If no srong facor srucure s presen n he error erms, hen he wo-pass procedure s asympocally equvalen o he nfeasble esmae λ and has asympoc varance Ω F. 4. Case wh weak observed facors bu no srong mssng facors Corollary Assume ha he sample conssng of {r, =,...,, =,..., } and {F, =,..., } comes from a daa-generang process ha sasfes he facor-prcng 5

17 model and assumpons FACORS, LOADIGS and ERRORS wh k here are weak observed facors and k v = 0 no mssng facor srucure n errors. hen he followng asympoc saemens hold smulaneously: λ P, λ = λ λ + AB + o p, λ P, λ = AB + o p, where AB AB p Γ + I k Σ u I k I k Σ u λ. 4 In he case when some of he observed facors have relavely small loadngs weak observed facors he wo-pass esmaor wll devae from he classcal case even f he dosyncrac errors are no srongly correlaed. he lm n 4 s non-random and s a non-zero vecor, and hus characerzes he asympoc bas. he wo-pass esmae λ P, of he rsk prema on weak facors F, s nconssen and converges n probably o an ncorrec value. he wo-pass esmae λ P, of rsk prema on srong facors F, s - conssen bu hs esmae has a bas of order, he same order of magnude as he sandard devaon of s asympocally gaussan dsrbuon. hs leads o confdence ses beng msplaced and sandard nferences on he rsk prema beng nvald. Inuon for he case wh weak observed facors and no facor srucure n he error erms. he resul of Corollary can be explaned n erms of classcal errorn-varables bas, or he so-called aenuaon bas. Indeed, he frs-pass esmae β of rsk exposure coeffcens β conans esmaon errors whch are sochascally of order O p / each: β = = F F = F r = β + u + o p, where he o p erm s relaed o he dfference beween Σ F = E[F EF F EF ] and F F. As a resul, he second-pass regresson encouners an error-n-varables problem. In he case of exposure o a srong observed facor, he esmaon error n β, s asympocally neglgble compared o he sze of he coeffcen β, self, and so hs esmaon error does no jeopardze conssency. However, he esmaon error n β, s asympocally of he same order of magnude as he coeffcen self. he frs-pass esmaon errors n β, behave lke a classcal measuremen error n he followng sense: he mposed assumpons guaranee ha he esmaon errors u, for dfferen asses 6

18 are asympocally uncorrelaed and ha hey are asympocally uncorrelaed wh β hemselves n he sense ha he sample correlaon beween β and u s asympocally neglgble. he bas we observe n Corollary s classc aenuaon bas, wh I k Σ u I k correspondng o he varance of he normalzed measuremen error u,. oe ha f Γ = Γ ββ s a block dagonal marx wh Γ β β = 0 k,k, he wo-pass procedure nferences abou λ wll no be dsurbed; namely, λ P, wll be -conssen and when mulpled by wll have an asympocally mean-zero Gaussan dsrbuon. he block-dagonaly assumpon, hough, s a very srong one: requres ha he values of β, be unrelaed o he values of β, for he same asse, whch s boh mplausble and no suppored n applcaons. For example, he sample correlaon coeffcen beween porfolos beas ha correspond o he marke porfolo and beas ha correspond o he SMB HML porfolo n he Fama French daase s equal o Case wh weak observed facors and srong mssng facors n errors Corollary 3 Assume ha he sample {r, =,...,, =,..., } and {F, =,..., } comes from a daa-generang process ha sasfes he facor-prcng model and assumpons FACORS, LOADIGS and ERRORS wh k here are weak observed facors and k v here s a mssng facor srucure n errors. hen he followng asympoc saemens hold smulaneously: λ P, λ = λ λ + AB + OV B + o p, λ P, λ = AB + OV B + o p, where OV B OV B AB AB I kβ ; ηγi kβ ; η + I k Σ u I k Γβµ + ηγ µµ η v η λ, I kβ ; ηγi kβ ; η + I k Σ u I k Ik Σ u λ. he dsrbuons of he rgh-hand-sde expressons are non-gaussan and are no cenered a zero. hs resul covers a more general case whch, as we argued before, s emprcally que relevan. Here some observed prcng facors may have relavely small loadngs We use he sascal erms correlaed and uncorrelaed nformally here, as he loadngs are no formally random varables. 7

19 weak facors, whle errors are hghly cross-seconally correlaed o he exen ha hey have srong mssng facor srucures. he wo-pass esmae λ P, of he rsk prema on weak facors F, s nconssen and, asympocally, has a poorly-cenered non-sandard dsrbuon. he wo-pass esmae λ P, of rsk prema on srong facors F, s - conssen bu hs esmae has a bas of order and an asympocally non-sandard dsrbuon. hs makes sandard nferences based on usual -sascs on he rsk prema nvald. Inuon for he case wh facor srucure n he error erms. In he presence of a srong facor srucure n he errors, frs-pass esmaes have he followng asympoc represenaon: β = = F F F r = = β + η µ + u + o p, 5 where η = = Σ F F v η. Agan, for he srong observed facors, he esmaon error n β, urns o be asympocally neglgble when compared o he szes of rsk exposure β, hemselves, whle he esmaon errors n β, whch are now equal o η µ / + u are of he sze O p /, whch s he same order of magnude as he β, s hemselves. he esmaon errors of β, dsor he asympocs and nvaldae classcal nferences. However, unlke he case covered by Corollary, he esmaon errors n hs seng do no behave lke classcal measuremen errors n wo respecs. Frs, he esmaon errors for dfferen asses are correlaed due o he presence of he common componen η n all of hem. Second, unless µ s cross-seconally uncorrelaed wh β so ha Γ βµ = 0 kf,k v, he esmaon error wll be correlaed wh s own regressor β. here s an addonal ssue worh nong wh he wo-pass procedure whch s classcally known as omed varable bas. Le us look a he second sep normalzed deal regresson whch we can oban by me-averagng equaon : r = λ β + η v, µ + e, 6 where η v, = = v η v 0 kv,, I kv. Here we nroduced normalzaon o make regresson 6 more conformable wh he classcal OLS seup. he regresson error erms e all have orders of magnude of O p, zero means and fne varances. Even hough n fne samples e may be weakly cross-seconally dependen, assumpon ERRORS guaranees ha hey are asympocally uncorrelaed. Imagne for a momen 8

20 ha we know β and µ for all asses. hen, regresson 6 wll ake he form of a classc OLS regresson, wh regressors β, and µ beng of order of magnude O, n he sense expressed n assumpon LOADIGS, 3 ha n he classcal regresson seng would lead o a -conssen and asympocally gaussan OLS esmaor of coeffcens on β s and µ s. he regressor β, s, n conras, of order O and carres a lo of nformaon, whch n he classcal regresson seng leads o an OLS esmaor of he coeffcen λ on hs regressor ha s super-conssen and asympocally cenered gaussan. However, because µ s unobserved, becomes a par of he error erm n he second-pass regresson, makng error erms cross-seconally correlaed; see, for example, Andrews 005 for a smlar phenomenon. A more classcal reference for hs phenomenon s an omed varable bas f Γ βµ 0 kf,k v, hen even f here were no frs-sage esmaon error and we knew β, runnng an OLS n a regresson of r on β would produce nvald resuls due o he omsson of µ. One queson ha may arse s wheher or no he omed varable bas s large. he answer o hs queson s closely relaed o he sze of he cross-seconal correlaon beween β and µ as expressed n Γ βµ. Unforunaely, here s no relable emprcal evdence on hs, as µ s unobserved and β s poorly esmaed and based n he drecon of µ see equaon 5. he problem wh esmaon of µ s ha he esmaor λ P, s nconssen, whch makes he resduals from he wo-pass procedure poor ndcaors of he rue errors, and esmang µ va he prncple componens analyss on he resduals does no produce good esmaes. However, even hough drec emprcal evdence on hs maer s absen, we have wo ndrec argumens whch sugges ha one should expec a hgh raher han low correlaon beween β and µ. One argumen s he emprcal observaon ha for many well-known facor-prcng models he esmaed beas for dfferen facors are exceponally hghly correlaed. Anoher argumen s relaed o our heorecal example, where he mssng facor srucure orgnaes as a resul of msmeasurng he rue rsk facor, and he sample correlaon beween β and µ equals. 5 ewly proposed esmaor 5. Idea of he proposed soluon he case wh no facor srucure n he error erms. We begn by solvng he easer case when no unobserved facor srucure s presen n he errors, whle some 3 hs s smlar o he assumpon from classcal regresson wh fxed regressors ha = x x A x, where x s h observaon for he regressor, and A x s a full rank fne marx. 9

21 observed facors are weak. As we dscussed before, n such a case he falure of he wopass procedure can be labeled a classcal measuremen error-n-varables problem, whch s ofen solved by fndng a proper nsrumen. Apparenly, s relavely easy o fnd a vald nsrumen n our seng f one s wllng o employ a sample-splng echnque. Le us dvde he se of me ndexes =,..., no wo non-nersecng equal subses and. I s more naural o make he frs half of he sample, and s second half; hen he procedure wll have greaer robusness as we dscuss below. Le us run he frs sep regresson wce separaely on each sub-sample: where β j = j F j F j j j F = F j j F, u j j F r = β + u j + o p, for j =,, = j j Σ j F F j F j e, and he o p erm s relaed o he dfference beween Σ F and j F j. he assumpon ERRORS guaranees ha he wo ses of esmaon uncerany, {u, =,..., } and {u, =,..., }, are ndependen condonally on F. In fac, he asympoc ndependence of he wo ses of errors wll hold more generally f one makes saonary assumpons and conrols he decay of me-seres dependence n errors e, and he sub-samples are formed o be frs and second halves of he sample, correspondngly. Gven he observaon abou ndependence of esmaon errors obaned from dfferen sub-samples, one may use an esmae of β from one sub-sample for example, β, as a regressor whle he oher n hs example, as an nsrumen. hs would represen a vald IV regresson. Indeed, he second sep regresson we run s: r = β λ β + e λ u. In hs regresson he regressor and he nsrumen are correlaed snce hey boh conan β, hence we have a relevance condon. he valdy condon holds for wo reasons: he par of he second-sep regresson error u s asympocally uncorrelaed wh he nsrumen β s asympocally un- ; assumpon ERRORS guaranees ha correlaed wh e. As we show below, hs procedure resores conssency and sandard nferences on he esmaes of rsk prema. Smlar deas, such as sample splng and jackknfe-ype esmaors, have been prevously employed n he leraure on many weak nsrumens e.g., Hansen, Hausman and ewey, 008. In ha leraure he erm many nsrumens s relaed o modelng he β 0

22 number of nsrumens as growng o nfny proporonally hough no always o he sample sze so-called dmensonaly asympocs, whle he erm weak appears due o a modelng assumpon ha makes he esmaon error of he reduced-form coeffcens be of he same order of magnude as he coeffcens hemselves so called local-o-zero asympocs. hs s parallel o he dmensonaly asympocs for a number of porfolos and he local-o-zero asympocs for rsk exposures of weak facors n our seup. In he many-weak-nsrumens seng, he regular SLS esmaor has a sgnfcan bas, and classcal nferences are asympocally nvald. ha problem can also be nerpreed as a classcal measuremen error-n-varables problem for he second sage regresson, where he regresson s run on he fed values from he frs-sage projecon of he orgnal regressor on he nsrumens. Some proposed soluons employ he second-sage nsrumenal varables regresson where, for each observaon, he regressor s obaned from a frs-sage regresson run on a sub-sample ha does no nclude ha observaon, and he orgnal nsrumen s sll used as an nsrumen Angrs, Imbens and Krueger, 999. hs makes he frs-sage error n he projecon uncorrelaed wh he nsrumen for hs specfc observaon. Conssency and classcal nferences are resored by sample-splng or leave-one-ou ype procedures. he case wh facor srucure n he error erms. As we dscussed before, he suaon n he model wh unobserved facor srucure has an addonal problem ha can be descrbed as he presence of omed and unobserved varable µ n regresson 6. However, afer examnng formula 5 for he frs-pass esmae we may noce ha we can oban a nosy proxy for µ f we ake he dfference beween wo esmaes for he same β obaned from dfferen sub-samples. Indeed, consder wo non-nersecng subses of me ndexes, and and assume hey have he same number, say τ, of me ndexes. hen β β = η τ η τ µ + u τ u. oce ha boh he coeffcen on µ and he nose erm u u are of he same order of magnude O p / τ. hs neher means ha he sgnal domnaes he nose, and hus we need a correcon o accoun for he nose, nor does he nose domnae he sgnal, and hus he proxy s no useless. Assume ha k v k F, whch mples ha we have a larger number of proxes han needed and we have a choce among hem. ow we assume ha we have a fxed and

23 full-rank k v k F marx A, and use A β β as he proxy. I s worh nong ha µ = τ Aη τ η τ A β β τ Aη τ η τ Au u, where η τ and η τ are asympocally ndependen non-degenerae k F k v gaussan assumng ha he sze of sub-samples τ ncreases o nfny wh ; he k v k v marx Aη τ η τ wll be nverble wh probably. he dea s o regress he average reurn r on β and A β β nsead of on unobserved β and µ. hs solves he omed varables par of he problem, bu he error-n-varables ssue sll remans. ha problem we solve va nsrumenal varables upon addonal sample splng. he ulmae dea goes as follows: spl he sample no four equal sub-samples along he me dmenson; calculae he frs-pass esmaes of rsk exposures for all four sub-samples; run an nsrumenal varables regresson usng A β 3 β as regressors and β and β 3 4 β as nsrumens. 5. Algorhm for consrucng a 4-spl esmaor Le us dvde he se of me ndexes no four equal non-nersecng subses j, j =,..., 4. For each asse and each subse j run a me-seres regresson o esmae he coeffcens of rsk exposure: β j = Run an IV regresson of r = wh nsrumens z = j F j F j j F j r. β and = r on regressors x β =, β β A β3, β 3 4, β where A s a non-random k v k F marx of rank k F. Denoe he SLS esmae of he coeffcen on regressor λ. 3 Repea sep hree more mes exchangng ndexes o 4 crcularly; ha s, he frs repeon s an IV regresson of r on regressors x β =, β 3 β A wh nsrumens z = by λ, ec. β4, β 4 4 Oban he 4-spl esmae as λ 4S = 4 4 j= λ j. β by ; β denoe he correspondng esmae

24 5 In order o compue an esmae of he covarance marx for λ 4S, denoe by X j he k marx of sacked regressors used n he IV regresson where λ j was obaned, and by Z j he k z marx of nsrumens from hs regresson here k z = k F s he number of nsrumens n a sngle regresson, and k = k F + k v s he number of regressors. Calculae G = G 0 k,k 0 k,k 0 k,k 0 k,k G 0 k,k 0 k,k 0 k,k 0 k,k G 3 0 k,k 0 k,k 0 k,k 0 k,k G 4, where G j = Xj P Z jx j, and P Z = Z Z Z Z. Also calculae Σ 0 = = z ϵ... z 4 ϵ 4 z ϵ... z 4 ϵ 4, where ϵ j s h resdual from he IV regresson where λ j was obaned, and z j = X j Z j Z j Z j z j. Also denoe R =,,, I 4 k F, whch s a 4k k F marx. Fnally, Σ 4S = R G Σ0 G R + Ω F, 0 kv,k F where Ω F s a conssen esmaor of he long-run varance of F. 5.3 Conssency of he 4-spl esmaor heorem Assume ha he samples {r, =,...,, =,... } and {F, =,..., } come from a daa-generang process ha sasfes he facor prcng model and assumpons FACORS, LOADIGS and ERRORS. Le boh and ncrease o nfny hen λ4s, λ = λ λ + O p / 0, Ω F, and mn{, } λ4s, λ = O p. Dscusson. λ 4S heorem esablshes he speed of conssency for he new 4-spl esmaor under exacly same assumpons used o show falure of he wo-pass esmaon procedure. he 4-spl esmaor for he rsk prema on he srong observed facor s 3

25 conssen, asympocally equvalen o λ and asympocally Gaussan, whle he 4-spl esmae of he rsk prema on he weak observed facor s conssen, and he speed of convergence depends on he relave sze of and. heorem shows ha he 4-spl esmaor has superor asympoc properes n comparson o he classcal wo-pass procedure for he rsk prema. 6 Inference procedures usng 4-spl esmaor heorem shows ha he new 4-spl esmaor s conssen bu does no provde a bass for sascal nference, namely, for confdence se consrucon or esng. In order o use heorem he researcher has o know whch observed facors are srong, and wh ha knowledge s/he can consruc a confdence se for he rsk prema on he srong observed facor only. However, n general here s no a pre-es ha successfully dscrmnaes beween weak and srong observed facors. he oher drawback of heorem s ha does no provde an asympoc dsrbuon for he esmaor of he rsk prema on a weak observed facor. Apparenly, he saed assumpons are no enough o oban he asympoc dsrbuon of he full 4-spl esmaor. We addonally need assumpons ha wll guaranee he valdy of some Cenra Lm heorems. Below we formulae he needed hgh-level assumpons and esablsh a resul abou sascal nferences usng he 4-spl esmaor. We also provde prmve assumpons ha wll guaranee ha our hgh-level assumpons wll hold n examples and dscuss how one can oban he needed Cenral Lm heorems. For a se of vecors a j we denoe a j 4 j= = a,..., a 4 as a long vecor conssng of he four vecors sacked upon each oher, smlarly for vecors a jj we denoe a jj j<j = a, a 3, a 4, a 3, a 4, a 34. Assumpon GAUSSIAIY = γ e γ u j 4 j= u j e u j 4 j= u j j<j Assume ha he followng convergence holds: ξ γe = ξ ξ = ξ γj 4 j= ξ ej 4, = j= ξ j,j j<j where ξ s a Gaussan vecor wh mean zero and covarance Σ ξ and assume ha ξ ξ p Σ ξ. = 4

26 heorem 3 Assume ha he samples {r, =,...,, =,... } and {F, =,..., } come from a daa-generang process sasfyng facor prcng model and assumpons FACORS, LOADIGS, ERRORS and GAUSSIAIY as boh and ncrease o nfny. hen Σ / 4S λ 4S λ 0, I k. heorem 3 suggess he use of and Wald sascs for he consrucon of confdence ses for he rsk prema as well as for esng hypoheses abou values of he rsk prema. hese nference procedures are very sandard ones and can be performed usng sandard economerc sofware. From a heorecal perspecve, however, he asympocs of he 4-spl esmaor are no fully sandard. echncally, he asympoc dsrbuon of he 4-spl esmaor s no Gaussan bu raher mxed Gaussan. hs s due o he fac ha he lm dsrbuon for hs esmaor can be wren as a Gaussan random vecor wh random varance. o undersand he nuon for why one ges an asympocally-random covarance marx one can look a equaon 6 and noce ha he coeffcen η v, on he omed varable µ s random, even asympocally. hs leads o a phenomenon where he amoun of nformaon conaned n he sample ha s used o correc he omed-varable problem wll be random as well, and hus, we have an asympocally-random covarance marx. heorem 3 shows ha a properly consruced proxy for he asympoc varance resores he asympoc gaussany of a muldmensonal -sasc even when he esmaor self s no asympocally gaussan. Anoher mporan aspec of heorem 3 s ha nferences or consrucon of he proxy for varance do no assume knowledge of he number or deny of srong/weak facors. hs s a desrable feaure, as we do no have a procedure ha can credbly dfferenae beween weak and srong facors. As prevously dscussed, even hough he man daa se conans observaons, he rsk prema canno be esmaed a a rae beer han. hs can be seen from equaon 3 as even f we know he rue values of β he regresson of r on β has a rue coeffcen equal o λ = λ+f EF. hs means ha he uncerany assocaed wh he devaons of = F from EF s unavodable. hs also jusfes he presence of he long-run varance of facors, Ω F, n he formula for varance Σ 4S. heorem also saes ha he dfference beween λ 4S and λ s of order. From he proof of heorem 3 we see ha hs dfference s mxed Gaussan, and he varance can be deduced from Σ IV. 5